Lagrange duality. The Lagrangian. We consider an optimization program of the form
|
|
- Berniece Rose
- 5 years ago
- Views:
Transcription
1 Lagrange duality Another way to arrive at the KKT conditions, and one which gives us some insight on solving constrained optimization problems, is through the Lagrange dual. The dual is a maximization program in λ, ν it is always concave (even when the original program is not convex), and gives us a systematic way to lower bound the optimal value. The Lagrangian We consider an optimization program of the form minimize f 0 (x) f m (x) 0, m = 1,..., M (1) x R N h p (x) = 0, p = 1,..., P. Much of what we will say below applies equally well to nonconvex programs as well as convex programs, so we will make it clear when we are taking the f m to be convex and the h p to be affine. We will take the domain of all of the f m and h p to be all of R N below; this just simplifies the exposition, we can easily replace this with the intersections of the dom f m and dom h p. We will assume that the intersection of the feasible set, C = {x : f m (x) 0, h p (x) = 0, m = 1,..., M, p = 1,..., P } is a non-empty and a subset R N. 1
2 The Lagrangian takes the constraints in the program above and integrates them into the objective function. The Lagrangian L : R N R M R P R associated with this optimization program is L(x, λ, ν) = f 0 (x) + λ m f m (x) + ν p h p (x) The x above are referred to as primal variables, and the λ, ν as either dual variables or Lagrange multipliers. The Lagrange dual function g(λ, ν) : R M R P R is the minimum of the Lagrangian over all values of x: ( ) g(λ, ν) = inf f 0 (x) + λ m f m (x) + ν p h p (x). x R N Since the dual is a pointwise infimum of a family of affine functions in λ, ν, g is concave regardless of whether or not the f m, h p are convex. The key fact about the dual function is that is it is everywhere a lower bound on the optimal value of the original program. If p is the optimal value for (1), then g(λ, ν) p, for all λ 0, ν R P. 2
3 This is (almost too) easy to see. For any feasible point x 0, λ m f m (x 0 ) + ν p h p (x 0 ) 0, and so L(x 0, λ, ν) f 0 (x 0 ), for all λ 0, ν R P, meaning g(λ, ν) = inf x R N L(x, λ, ν) L(x 0, λ, ν) f 0 (x 0 ). Since this holds for all feasible x 0, g(λ, ν) inf x C f 0 (x) = p. The (Lagrange) dual to the optimization program (1) is maximize g(λ, ν) subject to λ 0. (2) λ R M, ν R P The dual optimal value d is d = sup g(λ, ν) = sup λ 0, ν λ 0, ν inf L(x, λ, ν). x R N Since g(λ, ν) p, we know that d p. The quantity p d is called the duality gap. If p = d, then we say that (1) and (2) exhibit strong duality. 3
4 Certificates of (sub)optimality Any dual feasible 1 (λ, ν) gives us a lower bound on p, since g(λ, ν) p. If we have a primal feasible x, then we know that f 0 (x) p f 0 (x) g(λ, ν). We will refer to f 0 (x) g(λ, ν) as the duality gap for primal/dual (feasible) pair x, λ, ν. We know that p [g(λ, ν), f 0 (x)], and likewise d [g(λ, ν), f 0 (x)]. If we are ever able to reduce this gap to zero, then we know that x is primal optimal, and λ, ν are dual optimal. There are certain kinds of primal-dual algorithms that produce a series of (feasible) points x (k), λ (k), ν (k) at every iteration. We can then use f 0 (x (k) ) g(λ (k), ν (k) ) ɛ, as a stopping criteria, and know that our answer would yield an objective value no further than ɛ from optimal. Strong duality and the KKT conditions Suppose that for a convex program, the primal optimal value p an the dual optimal value d are equal p = d. 1 We simply need λ 0 for (λ, ν) to be dual feasible. 4
5 If x is a primal optimal point and λ, ν is a dual optimal point, then we must have f 0 (x ) = g(λ, ν ) ( = inf x R N f 0 (x ) + f 0 (x ). f 0 (x) + λ mf m (x) + λ mf m (x ) + ) νph p (x) νph p (x ) The last inequality follows from the fact that λ m 0 (dual feasibility), f m (x ) 0, and h p (x ) = 0 (primal feasibility). Since we started out and ended up with the same thing, all of the things above must be equal, and so λ mf m (x ) = 0, m = 1,..., M. Also, since we know x is a minimizer of L(x, λ, ν ) (second equality above), which is an unconstrained convex function (with λ, ν fixed), the gradient with respect to x must be zero: x L(x, λ, ν ) = f 0 (x )+ λ m f m (x )+ νp h p (x ) = 0. Thus strong duality immediately leads to the KKT conditions holding at the solution. Also, if you can find x, λ, ν that obey the KKT conditions, not only do you know that you have a primal optimal point on your hands, but also we have strong duality (and λ, ν are dual optimal). For if KKT holds, x L(x, λ, ν ) = 0, 5
6 meaning that x is a minimizer of L(x, λ, ν ), i.e. thus L(x, λ, ν ) L(x, λ, ν ), g(λ, ν ) = L(x, λ, ν ) = f 0 (x ) + λ mf m (x ) + = f 0 (x ), (by KKT), νph p (x ) and we have strong duality. The upshot of this is that the conditions for strong duality are essentially the same as those under which KKT is necessary. The program (1) and its dual (2) have strong duality if the f m are affine inequality constraints, or there is an x R N such that for all the f i which are not affine we have f i (x) < 0. 6
7 Examples 1. Inequality LP. Calculate the dual of minimize x, c subject to Ax b. x R N Answer: The Lagrangian is L(x, λ) = x, c + λ m ( x, a m b m ) = c T x λ T b + λ T Ax. This is a linear functional in x it is unbounded below unless c + A T λ = 0. Thus g(λ) = inf x ( ) c T x λ T b + λ T Ax = { λ, b, c + A T λ = 0, otherwise. So the Lagrange dual program is maximize λ, b subject to A T λ = c λ R M λ 0. 7
8 2. Standard form LP. Calculate the dual of minimize x R N x, c subject to Ax = b λ 0. 8
9 Least-squares. Calculate the dual of minimize x R N x 2 2 subject to Ax = b. Check that the duality gap is zero. Answer: maximize ν R M 1 4 νt AA T ν b T ν 9
10 3. Minimum norm. Calculate the dual of minimize x subject to Ax = b, x R N where is a general valid norm. Answer: Use f 0 (x) = x to ease notation below. We start with the Lagrangian: L(x, ν) = f 0 (x) + ν p ( x, a m b m ) = f 0 (x) ν, b + (A T ν) T x and so g(ν) = ν, b + inf x = ν, b sup x = ν, b f 0 ( A T ν), ( ) f 0 (x) + (A T ν) T x ( f 0 (x) (A T ν) T x ) where f 0 is the Fenchel dual of f 0 : f 0 (y) = sup( x, y f 0 (x)). x With f 0 =, we know already that so f 0 (y) = g(ν) = { 0, y 1,, otherwise, { ν, b, A T ν 1, otherwise. 10
11 Thus the dual program is maximize ν, b subject to A T ν 1, ν R P where is the dual norm of. 11
5. Duality. Lagrangian
5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized
More informationConvex Optimization M2
Convex Optimization M2 Lecture 3 A. d Aspremont. Convex Optimization M2. 1/49 Duality A. d Aspremont. Convex Optimization M2. 2/49 DMs DM par email: dm.daspremont@gmail.com A. d Aspremont. Convex Optimization
More informationConvex Optimization Boyd & Vandenberghe. 5. Duality
5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized
More informationExtreme Abridgment of Boyd and Vandenberghe s Convex Optimization
Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization Compiled by David Rosenberg Abstract Boyd and Vandenberghe s Convex Optimization book is very well-written and a pleasure to read. The
More informationLecture: Duality.
Lecture: Duality http://bicmr.pku.edu.cn/~wenzw/opt-2016-fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghe s lecture notes Introduction 2/35 Lagrange dual problem weak and strong
More information14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness.
CS/ECE/ISyE 524 Introduction to Optimization Spring 2016 17 14. Duality ˆ Upper and lower bounds ˆ General duality ˆ Constraint qualifications ˆ Counterexample ˆ Complementary slackness ˆ Examples ˆ Sensitivity
More informationI.3. LMI DUALITY. Didier HENRION EECI Graduate School on Control Supélec - Spring 2010
I.3. LMI DUALITY Didier HENRION henrion@laas.fr EECI Graduate School on Control Supélec - Spring 2010 Primal and dual For primal problem p = inf x g 0 (x) s.t. g i (x) 0 define Lagrangian L(x, z) = g 0
More informationLecture: Duality of LP, SOCP and SDP
1/33 Lecture: Duality of LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2017.html wenzw@pku.edu.cn Acknowledgement:
More informationLagrange Duality. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST)
Lagrange Duality Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline of Lecture Lagrangian Dual function Dual
More informationConvex Optimization & Lagrange Duality
Convex Optimization & Lagrange Duality Chee Wei Tan CS 8292 : Advanced Topics in Convex Optimization and its Applications Fall 2010 Outline Convex optimization Optimality condition Lagrange duality KKT
More informationICS-E4030 Kernel Methods in Machine Learning
ICS-E4030 Kernel Methods in Machine Learning Lecture 3: Convex optimization and duality Juho Rousu 28. September, 2016 Juho Rousu 28. September, 2016 1 / 38 Convex optimization Convex optimisation This
More informationEE/AA 578, Univ of Washington, Fall Duality
7. Duality EE/AA 578, Univ of Washington, Fall 2016 Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized
More informationConstrained Optimization and Lagrangian Duality
CIS 520: Machine Learning Oct 02, 2017 Constrained Optimization and Lagrangian Duality Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture. They may or may
More informationsubject to (x 2)(x 4) u,
Exercises Basic definitions 5.1 A simple example. Consider the optimization problem with variable x R. minimize x 2 + 1 subject to (x 2)(x 4) 0, (a) Analysis of primal problem. Give the feasible set, the
More informationOn the Method of Lagrange Multipliers
On the Method of Lagrange Multipliers Reza Nasiri Mahalati November 6, 2016 Most of what is in this note is taken from the Convex Optimization book by Stephen Boyd and Lieven Vandenberghe. This should
More informationDuality. Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities
Duality Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities Lagrangian Consider the optimization problem in standard form
More informationOptimization for Communications and Networks. Poompat Saengudomlert. Session 4 Duality and Lagrange Multipliers
Optimization for Communications and Networks Poompat Saengudomlert Session 4 Duality and Lagrange Multipliers P Saengudomlert (2015) Optimization Session 4 1 / 14 24 Dual Problems Consider a primal convex
More informationLagrangian Duality and Convex Optimization
Lagrangian Duality and Convex Optimization David Rosenberg New York University February 11, 2015 David Rosenberg (New York University) DS-GA 1003 February 11, 2015 1 / 24 Introduction Why Convex Optimization?
More informationOptimization for Machine Learning
Optimization for Machine Learning (Problems; Algorithms - A) SUVRIT SRA Massachusetts Institute of Technology PKU Summer School on Data Science (July 2017) Course materials http://suvrit.de/teaching.html
More informationTutorial on Convex Optimization: Part II
Tutorial on Convex Optimization: Part II Dr. Khaled Ardah Communications Research Laboratory TU Ilmenau Dec. 18, 2018 Outline Convex Optimization Review Lagrangian Duality Applications Optimal Power Allocation
More informationCS-E4830 Kernel Methods in Machine Learning
CS-E4830 Kernel Methods in Machine Learning Lecture 3: Convex optimization and duality Juho Rousu 27. September, 2017 Juho Rousu 27. September, 2017 1 / 45 Convex optimization Convex optimisation This
More informationLecture 18: Optimization Programming
Fall, 2016 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equality-constrained Optimization Inequality-constrained Optimization Mixture-constrained Optimization 3 Quadratic Programming
More informationConvex Optimization and Modeling
Convex Optimization and Modeling Duality Theory and Optimality Conditions 5th lecture, 12.05.2010 Jun.-Prof. Matthias Hein Program of today/next lecture Lagrangian and duality: the Lagrangian the dual
More informationShiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 4. Subgradient
Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 4 Subgradient Shiqian Ma, MAT-258A: Numerical Optimization 2 4.1. Subgradients definition subgradient calculus duality and optimality conditions Shiqian
More informationPrimal/Dual Decomposition Methods
Primal/Dual Decomposition Methods Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2018-19, HKUST, Hong Kong Outline of Lecture Subgradients
More informationEE364a Review Session 5
EE364a Review Session 5 EE364a Review announcements: homeworks 1 and 2 graded homework 4 solutions (check solution to additional problem 1) scpd phone-in office hours: tuesdays 6-7pm (650-723-1156) 1 Complementary
More informationHomework Set #6 - Solutions
EE 15 - Applications of Convex Optimization in Signal Processing and Communications Dr Andre Tkacenko JPL Third Term 11-1 Homework Set #6 - Solutions 1 a The feasible set is the interval [ 4] The unique
More informationInterior Point Algorithms for Constrained Convex Optimization
Interior Point Algorithms for Constrained Convex Optimization Chee Wei Tan CS 8292 : Advanced Topics in Convex Optimization and its Applications Fall 2010 Outline Inequality constrained minimization problems
More informationLagrangian Duality Theory
Lagrangian Duality Theory Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapter 14.1-4 1 Recall Primal and Dual
More informationLecture 7: Weak Duality
EE 227A: Conve Optimization and Applications February 7, 2012 Lecture 7: Weak Duality Lecturer: Laurent El Ghaoui 7.1 Lagrange Dual problem 7.1.1 Primal problem In this section, we consider a possibly
More informationConvex Optimization and SVM
Convex Optimization and SVM Problem 0. Cf lecture notes pages 12 to 18. Problem 1. (i) A slab is an intersection of two half spaces, hence convex. (ii) A wedge is an intersection of two half spaces, hence
More informationThe Lagrangian L : R d R m R r R is an (easier to optimize) lower bound on the original problem:
HT05: SC4 Statistical Data Mining and Machine Learning Dino Sejdinovic Department of Statistics Oxford Convex Optimization and slides based on Arthur Gretton s Advanced Topics in Machine Learning course
More informationEE 227A: Convex Optimization and Applications October 14, 2008
EE 227A: Convex Optimization and Applications October 14, 2008 Lecture 13: SDP Duality Lecturer: Laurent El Ghaoui Reading assignment: Chapter 5 of BV. 13.1 Direct approach 13.1.1 Primal problem Consider
More informationCSCI : Optimization and Control of Networks. Review on Convex Optimization
CSCI7000-016: Optimization and Control of Networks Review on Convex Optimization 1 Convex set S R n is convex if x,y S, λ,µ 0, λ+µ = 1 λx+µy S geometrically: x,y S line segment through x,y S examples (one
More informationA Brief Review on Convex Optimization
A Brief Review on Convex Optimization 1 Convex set S R n is convex if x,y S, λ,µ 0, λ+µ = 1 λx+µy S geometrically: x,y S line segment through x,y S examples (one convex, two nonconvex sets): A Brief Review
More informationConvex Optimization. Dani Yogatama. School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA. February 12, 2014
Convex Optimization Dani Yogatama School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA February 12, 2014 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12,
More informationMotivation. Lecture 2 Topics from Optimization and Duality. network utility maximization (NUM) problem:
CDS270 Maryam Fazel Lecture 2 Topics from Optimization and Duality Motivation network utility maximization (NUM) problem: consider a network with S sources (users), each sending one flow at rate x s, through
More information12. Interior-point methods
12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
More informationHW1 solutions. 1. α Ef(x) β, where Ef(x) is the expected value of f(x), i.e., Ef(x) = n. i=1 p if(a i ). (The function f : R R is given.
HW1 solutions Exercise 1 (Some sets of probability distributions.) Let x be a real-valued random variable with Prob(x = a i ) = p i, i = 1,..., n, where a 1 < a 2 < < a n. Of course p R n lies in the standard
More informationApplications of Linear Programming
Applications of Linear Programming lecturer: András London University of Szeged Institute of Informatics Department of Computational Optimization Lecture 9 Non-linear programming In case of LP, the goal
More informationLagrange Relaxation and Duality
Lagrange Relaxation and Duality As we have already known, constrained optimization problems are harder to solve than unconstrained problems. By relaxation we can solve a more difficult problem by a simpler
More informationConstrained optimization
Constrained optimization DS-GA 1013 / MATH-GA 2824 Optimization-based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_fall17/index.html Carlos Fernandez-Granda Compressed sensing Convex constrained
More informationECE Optimization for wireless networks Final. minimize f o (x) s.t. Ax = b,
ECE 788 - Optimization for wireless networks Final Please provide clear and complete answers. PART I: Questions - Q.. Discuss an iterative algorithm that converges to the solution of the problem minimize
More informationConvex Optimization. Newton s method. ENSAE: Optimisation 1/44
Convex Optimization Newton s method ENSAE: Optimisation 1/44 Unconstrained minimization minimize f(x) f convex, twice continuously differentiable (hence dom f open) we assume optimal value p = inf x f(x)
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EECS 227A Nonlinear and Convex Optimization. Solutions 5 Fall 2009
UC Berkeley Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Solutions 5 Fall 2009 Reading: Boyd and Vandenberghe, Chapter 5 Solution 5.1 Note that
More informationConvex Optimization. Lecture 12 - Equality Constrained Optimization. Instructor: Yuanzhang Xiao. Fall University of Hawaii at Manoa
Convex Optimization Lecture 12 - Equality Constrained Optimization Instructor: Yuanzhang Xiao University of Hawaii at Manoa Fall 2017 1 / 19 Today s Lecture 1 Basic Concepts 2 for Equality Constrained
More informationLinear and Combinatorial Optimization
Linear and Combinatorial Optimization The dual of an LP-problem. Connections between primal and dual. Duality theorems and complementary slack. Philipp Birken (Ctr. for the Math. Sc.) Lecture 3: Duality
More informationSubgradient. Acknowledgement: this slides is based on Prof. Lieven Vandenberghes lecture notes. definition. subgradient calculus
1/41 Subgradient Acknowledgement: this slides is based on Prof. Lieven Vandenberghes lecture notes definition subgradient calculus duality and optimality conditions directional derivative Basic inequality
More informationNumerical Optimization
Linear Programming - Interior Point Methods Computer Science and Automation Indian Institute of Science Bangalore 560 012, India. NPTEL Course on Example 1 Computational Complexity of Simplex Algorithm
More informationA Tutorial on Convex Optimization II: Duality and Interior Point Methods
A Tutorial on Convex Optimization II: Duality and Interior Point Methods Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California 94304 email: hhindi@parc.com Abstract In recent years, convex
More informationLecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem
Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem Michael Patriksson 0-0 The Relaxation Theorem 1 Problem: find f := infimum f(x), x subject to x S, (1a) (1b) where f : R n R
More informationLagrange Relaxation: Introduction and Applications
1 / 23 Lagrange Relaxation: Introduction and Applications Operations Research Anthony Papavasiliou 2 / 23 Contents 1 Context 2 Applications Application in Stochastic Programming Unit Commitment 3 / 23
More information4. Algebra and Duality
4-1 Algebra and Duality P. Parrilo and S. Lall, CDC 2003 2003.12.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone
More informationminimize x subject to (x 2)(x 4) u,
Math 6366/6367: Optimization and Variational Methods Sample Preliminary Exam Questions 1. Suppose that f : [, L] R is a C 2 -function with f () on (, L) and that you have explicit formulae for
More information10 Numerical methods for constrained problems
10 Numerical methods for constrained problems min s.t. f(x) h(x) = 0 (l), g(x) 0 (m), x X The algorithms can be roughly divided the following way: ˆ primal methods: find descent direction keeping inside
More informationIntroduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.1-4.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation Instructor: Moritz Hardt Email: hardt+ee227c@berkeley.edu Graduate Instructor: Max Simchowitz Email: msimchow+ee227c@berkeley.edu
More informationThe Karush-Kuhn-Tucker (KKT) conditions
The Karush-Kuhn-Tucker (KKT) conditions In this section, we will give a set of sufficient (and at most times necessary) conditions for a x to be the solution of a given convex optimization problem. These
More informationLecture 9 Sequential unconstrained minimization
S. Boyd EE364 Lecture 9 Sequential unconstrained minimization brief history of SUMT & IP methods logarithmic barrier function central path UMT & SUMT complexity analysis feasibility phase generalized inequalities
More informationTutorial on Convex Optimization for Engineers Part II
Tutorial on Convex Optimization for Engineers Part II M.Sc. Jens Steinwandt Communications Research Laboratory Ilmenau University of Technology PO Box 100565 D-98684 Ilmenau, Germany jens.steinwandt@tu-ilmenau.de
More informationConvex Optimization and Support Vector Machine
Convex Optimization and Support Vector Machine Problem 0. Consider a two-class classification problem. The training data is L n = {(x 1, t 1 ),..., (x n, t n )}, where each t i { 1, 1} and x i R p. We
More informationLagrangian Duality. Richard Lusby. Department of Management Engineering Technical University of Denmark
Lagrangian Duality Richard Lusby Department of Management Engineering Technical University of Denmark Today s Topics (jg Lagrange Multipliers Lagrangian Relaxation Lagrangian Duality R Lusby (42111) Lagrangian
More informationOptimality Conditions for Constrained Optimization
72 CHAPTER 7 Optimality Conditions for Constrained Optimization 1. First Order Conditions In this section we consider first order optimality conditions for the constrained problem P : minimize f 0 (x)
More informationSupport Vector Machines
Support Vector Machines Support vector machines (SVMs) are one of the central concepts in all of machine learning. They are simply a combination of two ideas: linear classification via maximum (or optimal
More informationLecture 7: Convex Optimizations
Lecture 7: Convex Optimizations Radu Balan, David Levermore March 29, 2018 Convex Sets. Convex Functions A set S R n is called a convex set if for any points x, y S the line segment [x, y] := {tx + (1
More informationQuiz Discussion. IE417: Nonlinear Programming: Lecture 12. Motivation. Why do we care? Jeff Linderoth. 16th March 2006
Quiz Discussion IE417: Nonlinear Programming: Lecture 12 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 16th March 2006 Motivation Why do we care? We are interested in
More informationAdditional Homework Problems
Additional Homework Problems Robert M. Freund April, 2004 2004 Massachusetts Institute of Technology. 1 2 1 Exercises 1. Let IR n + denote the nonnegative orthant, namely IR + n = {x IR n x j ( ) 0,j =1,...,n}.
More informationLecture 8. Strong Duality Results. September 22, 2008
Strong Duality Results September 22, 2008 Outline Lecture 8 Slater Condition and its Variations Convex Objective with Linear Inequality Constraints Quadratic Objective over Quadratic Constraints Representation
More informationExample Problem. Linear Program (standard form) CSCI5654 (Linear Programming, Fall 2013) Lecture-7. Duality
CSCI5654 (Linear Programming, Fall 013) Lecture-7 Duality Lecture 7 Slide# 1 Lecture 7 Slide# Linear Program (standard form) Example Problem maximize c 1 x 1 + + c n x n s.t. a j1 x 1 + + a jn x n b j
More informationNonlinear Optimization: What s important?
Nonlinear Optimization: What s important? Julian Hall 10th May 2012 Convexity: convex problems A local minimizer is a global minimizer A solution of f (x) = 0 (stationary point) is a minimizer A global
More informationDuality. Geoff Gordon & Ryan Tibshirani Optimization /
Duality Geoff Gordon & Ryan Tibshirani Optimization 10-725 / 36-725 1 Duality in linear programs Suppose we want to find lower bound on the optimal value in our convex problem, B min x C f(x) E.g., consider
More informationConvex Optimization. Ofer Meshi. Lecture 6: Lower Bounds Constrained Optimization
Convex Optimization Ofer Meshi Lecture 6: Lower Bounds Constrained Optimization Lower Bounds Some upper bounds: #iter μ 2 M #iter 2 M #iter L L μ 2 Oracle/ops GD κ log 1/ε M x # ε L # x # L # ε # με f
More informationIntroduction to Machine Learning Lecture 7. Mehryar Mohri Courant Institute and Google Research
Introduction to Machine Learning Lecture 7 Mehryar Mohri Courant Institute and Google Research mohri@cims.nyu.edu Convex Optimization Differentiation Definition: let f : X R N R be a differentiable function,
More informationLecture 6: Conic Optimization September 8
IE 598: Big Data Optimization Fall 2016 Lecture 6: Conic Optimization September 8 Lecturer: Niao He Scriber: Juan Xu Overview In this lecture, we finish up our previous discussion on optimality conditions
More informationNumerical Optimization
Constrained Optimization Computer Science and Automation Indian Institute of Science Bangalore 560 012, India. NPTEL Course on Constrained Optimization Constrained Optimization Problem: min h j (x) 0,
More informationPrimal-dual Subgradient Method for Convex Problems with Functional Constraints
Primal-dual Subgradient Method for Convex Problems with Functional Constraints Yurii Nesterov, CORE/INMA (UCL) Workshop on embedded optimization EMBOPT2014 September 9, 2014 (Lucca) Yu. Nesterov Primal-dual
More informationSolution to EE 617 Mid-Term Exam, Fall November 2, 2017
Solution to EE 67 Mid-erm Exam, Fall 207 November 2, 207 EE 67 Solution to Mid-erm Exam - Page 2 of 2 November 2, 207 (4 points) Convex sets (a) (2 points) Consider the set { } a R k p(0) =, p(t) for t
More information4TE3/6TE3. Algorithms for. Continuous Optimization
4TE3/6TE3 Algorithms for Continuous Optimization (Duality in Nonlinear Optimization ) Tamás TERLAKY Computing and Software McMaster University Hamilton, January 2004 terlaky@mcmaster.ca Tel: 27780 Optimality
More informationSupport Vector Machines: Maximum Margin Classifiers
Support Vector Machines: Maximum Margin Classifiers Machine Learning and Pattern Recognition: September 16, 2008 Piotr Mirowski Based on slides by Sumit Chopra and Fu-Jie Huang 1 Outline What is behind
More information10-725/ Optimization Midterm Exam
10-725/36-725 Optimization Midterm Exam November 6, 2012 NAME: ANDREW ID: Instructions: This exam is 1hr 20mins long Except for a single two-sided sheet of notes, no other material or discussion is permitted
More informationEquality constrained minimization
Chapter 10 Equality constrained minimization 10.1 Equality constrained minimization problems In this chapter we describe methods for solving a convex optimization problem with equality constraints, minimize
More informationGeneralization to inequality constrained problem. Maximize
Lecture 11. 26 September 2006 Review of Lecture #10: Second order optimality conditions necessary condition, sufficient condition. If the necessary condition is violated the point cannot be a local minimum
More informationPart IB Optimisation
Part IB Optimisation Theorems Based on lectures by F. A. Fischer Notes taken by Dexter Chua Easter 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationOptimality, Duality, Complementarity for Constrained Optimization
Optimality, Duality, Complementarity for Constrained Optimization Stephen Wright University of Wisconsin-Madison May 2014 Wright (UW-Madison) Optimality, Duality, Complementarity May 2014 1 / 41 Linear
More informationLECTURE 25: REVIEW/EPILOGUE LECTURE OUTLINE
LECTURE 25: REVIEW/EPILOGUE LECTURE OUTLINE CONVEX ANALYSIS AND DUALITY Basic concepts of convex analysis Basic concepts of convex optimization Geometric duality framework - MC/MC Constrained optimization
More informationLecture Note 5: Semidefinite Programming for Stability Analysis
ECE7850: Hybrid Systems:Theory and Applications Lecture Note 5: Semidefinite Programming for Stability Analysis Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio State
More informationSECTION C: CONTINUOUS OPTIMISATION LECTURE 11: THE METHOD OF LAGRANGE MULTIPLIERS
SECTION C: CONTINUOUS OPTIMISATION LECTURE : THE METHOD OF LAGRANGE MULTIPLIERS HONOUR SCHOOL OF MATHEMATICS OXFORD UNIVERSITY HILARY TERM 005 DR RAPHAEL HAUSER. Examples. In this lecture we will take
More informationKarush-Kuhn-Tucker Conditions. Lecturer: Ryan Tibshirani Convex Optimization /36-725
Karush-Kuhn-Tucker Conditions Lecturer: Ryan Tibshirani Convex Optimization 10-725/36-725 1 Given a minimization problem Last time: duality min x subject to f(x) h i (x) 0, i = 1,... m l j (x) = 0, j =
More informationE5295/5B5749 Convex optimization with engineering applications. Lecture 5. Convex programming and semidefinite programming
E5295/5B5749 Convex optimization with engineering applications Lecture 5 Convex programming and semidefinite programming A. Forsgren, KTH 1 Lecture 5 Convex optimization 2006/2007 Convex quadratic program
More informationCSCI 1951-G Optimization Methods in Finance Part 09: Interior Point Methods
CSCI 1951-G Optimization Methods in Finance Part 09: Interior Point Methods March 23, 2018 1 / 35 This material is covered in S. Boyd, L. Vandenberge s book Convex Optimization https://web.stanford.edu/~boyd/cvxbook/.
More informationConvex Optimization Lecture 13
Convex Optimization Lecture 13 Today: Interior-Point (continued) Central Path method for SDP Feasibility and Phase I Methods From Central Path to Primal/Dual Central'Path'Log'Barrier'Method Init: Feasible&#
More informationPrimal-Dual Interior-Point Methods for Linear Programming based on Newton s Method
Primal-Dual Interior-Point Methods for Linear Programming based on Newton s Method Robert M. Freund March, 2004 2004 Massachusetts Institute of Technology. The Problem The logarithmic barrier approach
More informationCOM S 578X: Optimization for Machine Learning
COM S 578X: Optimization for Machine Learning Lecture Note 4: Duality Jia (Kevin) Liu Assistant Professor Department of Computer Science Iowa State University, Ames, Iowa, USA Fall 2018 JKL (CS@ISU) COM
More informationExample: feasibility. Interpretation as formal proof. Example: linear inequalities and Farkas lemma
4-1 Algebra and Duality P. Parrilo and S. Lall 2006.06.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone of valid
More informationDuality Uses and Correspondences. Ryan Tibshirani Convex Optimization
Duality Uses and Correspondences Ryan Tibshirani Conve Optimization 10-725 Recall that for the problem Last time: KKT conditions subject to f() h i () 0, i = 1,... m l j () = 0, j = 1,... r the KKT conditions
More informationDuality in Linear Programs. Lecturer: Ryan Tibshirani Convex Optimization /36-725
Duality in Linear Programs Lecturer: Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: proximal gradient descent Consider the problem x g(x) + h(x) with g, h convex, g differentiable, and
More informationLecture 8 Plus properties, merit functions and gap functions. September 28, 2008
Lecture 8 Plus properties, merit functions and gap functions September 28, 2008 Outline Plus-properties and F-uniqueness Equation reformulations of VI/CPs Merit functions Gap merit functions FP-I book:
More informationDuality (Continued) min f ( x), X R R. Recall, the general primal problem is. The Lagrangian is a function. defined by
Duality (Continued) Recall, the general primal problem is min f ( x), xx g( x) 0 n m where X R, f : X R, g : XR ( X). he Lagrangian is a function L: XR R m defined by L( xλ, ) f ( x) λ g( x) Duality (Continued)
More informationMath 273a: Optimization Subgradients of convex functions
Math 273a: Optimization Subgradients of convex functions Made by: Damek Davis Edited by Wotao Yin Department of Mathematics, UCLA Fall 2015 online discussions on piazza.com 1 / 42 Subgradients Assumptions
More informationLagrangian Duality. Evelien van der Hurk. DTU Management Engineering
Lagrangian Duality Evelien van der Hurk DTU Management Engineering Topics Lagrange Multipliers Lagrangian Relaxation Lagrangian Duality 2 DTU Management Engineering 42111: Static and Dynamic Optimization
More information